### Abstract

Variational principles are proved for self-adjoint operator functions arising from variational evolution equations of the form ⟨z''(t),y⟩ + d[z'(t), y] +a0[z(t), y] = 0. Here a0 and d are densely defined, symmetric and positive sesquilinear forms on a Hilbert space H. We associate with the variational evolution equation an equivalent Cauchy problem corresponding to a block operator matrix A , the forms t(λ)[x, y] := λ2⟨x, y⟩ + λd[x, y] +a0[x, y], where λ ∈ ℂ and x, y are in the domain of the form a0, and a corresponding operator family T(λ). Using form methods we define a generalized Rayleigh functional and characterize the eigenvalues above the essential spectrum of A by a min-max and a max-min variational principle. The obtained results are illustrated with a damped beam equation.

Original language | English |
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Pages (from-to) | 501-531 |

Number of pages | 31 |

Journal | Operators and Matrices |

Volume | 10 |

Issue number | 3 |

DOIs | |

Publication status | Published - 30 Sep 2016 |

### Keywords

- block operator matrices
- variational principle
- operator function
- second-order equation
- spectrum
- essential spectrum
- sectorial form

## Cite this

Jacob, B., Langer, M., & Trunk, C. (2016). Variational principles for self-adjoint operator functions arising from second order systems.

*Operators and Matrices*,*10*(3), 501-531. https://doi.org/10.7153/oam-10-29